L2-Betti numbers arising from the lamplighter group

被引:0
|
作者
Ara, Pere [1 ]
Claramunt, Joan [2 ]
机构
[1] Univ Autonoma Barcelona, Dept Matemat, Edifici Cc, Barcelona 08193, Spain
[2] Univ Lancaster, Dept Math & Stat, Lancaster LA1 4YW, England
基金
欧洲研究理事会;
关键词
l(2)-Betti number; Atiyah conjecture; Rank function; Lamplighter; Odometer; GROUP-RING; ATIYAH; ELEMENTS;
D O I
10.1007/s10801-021-01044-8
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute l(2)-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, we obtain a whole family of irrational l(2)-Betti numbers arising from the lamplighter group algebra Q[Z(2) (sic) Z]. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational l(2)-Betti numbers from the algebras Q[Z(n) (sic) Z], where n >= 2 is a natural number. We also apply the techniques developed to the generalized odometer algebra (O)((n) over bar), where (n) over bar is a supernatural number. We compute its *-regular closure, and this allows us to fully characterize the set of O((n) over bar)-Betti numbers.
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页码:1201 / 1245
页数:45
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